The Minkowski vacuum model with Lambda = 0: field law, free-fall kinematics and clock readings

(The model and its field law.) By The flat model spacetime: the Minkowski metric on R^4 and its inertial chart clauses (a)-(c), $(M,g,N)$ is a smooth time-oriented Lorentzian four-manifold without boundary; by [A1] it satisfies the spacetime postulate, and $T=0$ is a smooth symmetric covariant $2$-tensor field on the matter domain $M$. By [L1], $G_{ab}\equiv 0$; with $\mathrm{\Lambda }=0$ and $T=0$, $G_{ab}+\mathrm{\Lambda }{g}_{ab}=0=\kappa T_{ab}$ holds identically, and ${\mathrm{\nabla }}^a{T}_{ab}=0$ trivially, so the prescription is admissible by [A1] and the field law holds by [A2]; this is also the flat vacuum solution of [L2]. This proves (a).

(Free fall and clock readings.) Let $p,U_0$ be as in (b), and write $U_0=cv$ with $\eta (v,v)=-1$, $v^0>0$; the normalization is possible by The flat model spacetime: the Minkowski metric on R^4 and its inertial chart clause (c). By [A3] the physical worldline is a restriction of the unique maximal future timelike geodesic through $(p,U_0)$, with proper time as affine parameter. By [L1] this maximal mathematical geodesic is the straight line $x(\gamma (s))=x_0+sv$. With proper-time parametrization $\tau =s\mathrm{/}c$ (the normalization $\lambda =1$ of [L1]), this is $x(\gamma (\tau ))=x_0+c\tau v$; the coordinate velocity is the constant $w=(c\mathrm{/}{v}^0)(v^1,v^2,v^3)$ with $\mathrm{\mid }w\mathrm{\mid }<c$ by [L1]. Finally, for two coordinate times in the physical domain, the ideal clock reads proper time by [A3], and from the parametrization $t=x^0\mathrm{/}c$ along the line, $\mathrm{\Delta }t=v^0\mathrm{\Delta }\tau$; since $\mathrm{\mid }w{\mathrm{\mid }}^2\mathrm{/}{c}^2=1-1\mathrm{/}(v^0{)}^2$ by [L1] clause (e), $\sqrt{1-\mathrm{\mid }w{\mathrm{\mid }}^2\mathrm{/}{c}^2}=1\mathrm{/}{v}^0$ and hence $\mathrm{\Delta }\tau =(1\mathrm{/}{v}^0)(t_1-t_0)=\sqrt{1-\mathrm{\mid }w{\mathrm{\mid }}^2\mathrm{/}{c}^2}(t_1-t_0)$. This proves (b).

(No tidal response.) The curvature vanishes identically by [L1], so by [L3] every free-fall congruence satisfies $D_{\tau }^{2}J=0$. This proves (c).

(Another flat vacuum model.) Fix a length $\mathrm{\ell }>0$. Let the discrete zero-dimensional Lie group $\mathbb{Z}$ act on ${\mathbb{R}}^4$ by $n\cdot z=z+(n\mathrm{\ell },0,0,0)$. Translations are smooth; the action is free since $n\mathrm{\ell }=0$ forces $n=0$. For any compact set $A\subset {\mathbb{R}}^4\times {\mathbb{R}}^4$, both projections have bounded time coordinates, so its inverse image under $(n,z)\mapsto (n\cdot z,z)$ has only finitely many integer components. Each component is closed in a compact projection of $A$, hence compact; the inverse image is a finite union of compact sets. The action is therefore proper. By [F1], $Q:={\mathbb{R}}^4\mathrm{/}\mathbb{Z}$ is a smooth four-manifold with a locally diffeomorphic projection $q$. Define $g_Q$ and $N_Q$ by $q^{\ast }{g}_Q=\eta$ and $(N_Q{)}_{q(z)}=d{q}_z({\mathrm{\partial }}_0)$; these are well defined because translations preserve $\eta$ and ${\mathrm{\partial }}_0$. In each inverse chart of $q$ they are smooth with $g_Q$ having constant components $\eta$ and $g_Q(N_Q,N_Q)=-1$. Thus $Q$ is time-oriented and flat. With $T_Q=0$, $\mathrm{\Lambda }=0$, its Einstein tensor and stress divergence vanish, so it satisfies the same adopted vacuum field law and matter consistency condition.

(The quotient has different smooth global topology.) The translation-invariant covector $d{z}^0$ descends to a smooth covector field $\alpha$ on $Q$, with $q^{\ast }\alpha =d{z}^0$. Its coordinate components satisfy ${\mathrm{\partial }}_i{\alpha }_j={\mathrm{\partial }}_j{\alpha }_i$ in every chart: in inverse charts they are constant, and the equality is preserved by coordinate changes by the chain rule and commuting mixed partials. The closed curve $\sigma (s)=q(s,0,0,0)$, $0\le s\le \mathrm{\ell }$, satisfies ${\int }_0^{\mathrm{\ell }}{\alpha }_{\sigma (s)}({\sigma }^{\mathrm{\prime }}(s))ds=\mathrm{\ell }$. If a diffeomorphism $F:{\mathbb{R}}^4\to Q$ existed, write $F^{\ast }\alpha =a_i(z)d{z}^i$; the same component symmetry holds for $a$. Define $f(z)={\int }_0^1{a}_i(tz)z^{i}dt$. By [F2], ${\mathrm{\partial }}_{j}f(z)={\int }_0^1(t{z}^i{\mathrm{\partial }}_j{a}_i(tz)+a_j(tz))dt={\int }_0^1\frac{d}{dt}(t{a}_j(tz))dt=a_j(z)$. Hence $\alpha =d(f\circ F^{-1})$. Its integral around $\sigma$ would then be the endpoint difference of a single-valued smooth function and equal zero, contradicting $\mathrm{\ell }>0$. Thus $Q$ is not diffeomorphic to ${\mathbb{R}}^4$, proving the different-topology assertion of (d).

(A non-flat vacuum solution.) On ${\mathbb{R}}^4$ with length coordinates $(u,v,x,y)$ put $H=(x^2-y^2)\mathrm{/}{\mathrm{\ell }}^2$ and $g_H=-du\otimes dv-dv\otimes du+dx\otimes dx+dy\otimes dy+Hdu\otimes du$. The $(u,v)$ block has determinant $-1$ and therefore one positive and one negative eigenvalue; the other two directions are positive. This is a smooth Lorentzian metric, time-oriented by $N_H={\mathrm{\partial }}_u+(H+1){\mathrm{\partial }}_v\mathrm{/}2$, since $g_H(N_H,N_H)=-1$. Its inverse has $g_H^{uv}=g_H^{vu}=-1$, $g_H^{vv}=-H$, $g_H^{xx}=g_H^{yy}=1$, and all other entries zero. Substitution into the Christoffel formula of [F2] gives only ${\mathrm{\Gamma }}^v{}_{ui}={\mathrm{\Gamma }}^v{}_{iu}=-{\mathrm{\partial }}_{i}H\mathrm{/}2$ and ${\mathrm{\Gamma }}^i{}_{uu}=-{\mathrm{\partial }}_{i}H\mathrm{/}2$ for $i=x,y$. The traces ${\mathrm{\Gamma }}^a{}_{ab}$ vanish, and the product terms in the Ricci contraction vanish: every nonzero symbol has a lower $u$ and either upper $v$ or two lower $u$ with transverse upper index, whereas every symbol with lower $v$ or upper $u$ is zero. Thus the only possible nonzero Ricci component is ${\mathrm{Ric}}_{uu}=-(H_{xx}+H_{yy})\mathrm{/}2=0$; nevertheless $R^x{}_{uxu}=-H_{xx}\mathrm{/}2=-1\mathrm{/}{\mathrm{\ell }}^2\mathrm{\ne }0$. With $T_H=0$, $\mathrm{\Lambda }=0$ this gives another admissible vacuum solution of the same field law, now non-flat.

(Conclusion and status.) Steps 1.1-1.3 establish (a)-(c) for the stated Minkowski model; steps 1.4, 2.1 and 2.2 establish the comparison existence claims of (d). All use the adopted postulates and explicit mathematics, with no observation, apparatus or measured input. Neither these deductions nor the comparison constructions assert that any of the models is realized in nature.